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markdow's avatar

Thue-Morse in rabbit land

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"So natralists observe, a flea
Hath smaller fleas that on him prey,
And these have smaller fleas that bite 'em,
And so proceed
ad infinitum." Jonathan Swift

"Great fleas have little fleas upon their backs to bite 'em,
And little fleas have lesser fleas, and so ad infinitum.
And the great fleas themselves, in turn, have greater fleas to go on,
While these again have greater still, and greater still, and so on."
Augustus De Morgan


An infinite dissection of the plane by logarithmic spirals.

There is a nice bistable perception in the animation: when I first see it, I percieve a wholistic pattern rotating clockwise and shrinking. But after a short time, I see half the contours as fixed, and the other half rotating (with no shrinking). Try fixating on a single edge point to experience the second percept.

The pattern is the basis of Orthologia trispiralis, and the pattern and transform is the basis for the animation Orthologia twist.

Here's the geeky description. It's a checkerboard division (8x2) of the plane by logarithmic spirals with a pitch of +/- pi/4 radians. The checkerboard is colored (orthogonal to the contours) by the first 2^3 = 8 elements of the Thue-Morse sequence, [0 1 1 0 1 0 0 1]. The relative phase between the two sets of opposite chirality spirals rotates through pi/2 radians; in this example the phase of one set is fixed.

Matlab

[Broken link: A volume rendering derived from the animation, with the time axis transposed to a third spatial dimension.]


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Seaxwulf's avatar
I'll not betray my ignorance by going to deep into the mathematical aspect of this, which is of course remarkably interesting and inscrutably beyond my ken. I will, however, submit that I have followed your electronic paper-trail in hopes of finding more biographical data concerning the progenitors of this Thue-Morse connection. Specifically, I am fixing to locate information regarding the Morse component. So, where you have primacy of association with this concept it is my hope you might enlighten me, as it goes.
markdow's avatar
A brief summary at en.wikipedia.org/wiki/Thue%E2%…:

"The Thue–Morse sequence was first studied by Eugène Prouhet in 1851, who applied it to number theory. However, Prouhet did not mention the sequence explicitly; this was left to Axel Thue in 1906, who used it to found the study of combinatorics on words. The sequence was only brought to worldwide attention with the work of Marston Morse in 1921, when he applied it to differential geometry."

So Prouhet looked at it, Thue formally defined it and found some pure math applications, and Morse showed it applied to an area of mathematical physics (and therefore modern physics).

History and naming conventions leave out many between. It is a nice short idea, for which a mathematical background is not needed except for putting it in its place within formal systems.
Seaxwulf's avatar
Yes. I did read that, thank you. No further biographical details on Morse, then? That's a pity! Thank you, though, for taking the time to help me out!
markdow's avatar
I just found this biographical summary of Marston Morse:

Seaxwulf's avatar

Oh, now thanks so much! See, I happen to belong to the Morse Clan, as it goes, and it's always fascinating to see others of the name floating around. We aren't a well-connected family and seem to find ourselves where we least expect it. Thanks again, at any rate! All the best to you and your endeavours!

markdow's avatar
Also see my reply to myself on the 'T-M in RL' page.
markdow's avatar
A bit more, with original reference, from The ubiquitous Prouhet-Thue-Morse sequence, John-Paull Allouche and Jeffrey Shallit:

Morse rediscovered the sequence in connection with differential geometry. He proved the following:

"On a surface of negative curvature having at least two different normal segments there exists a set of geodesics that are recurrent without being periodic and this set has the power of the continuum."

M. Morse, Recurrent geodesics on a surface of negative curvature, Trans. Amer. Math. Soc. 22 (1921), 84-100.

[This was soon after General Relativity showed that the shape of the structure of our space was not trivial and had a direct bearing on physics.]

[I love that phrase, "power of the continuum".]
vidthekid's avatar
If I'd thought of this, I'd have used a simple alternating sequence instead of that Thue-Morse thing. But now I see that the apparent uneven spacing of the spirals resulting from that sequence gives the piece an overall more dynamic appearance.
Pinto-Warrior's avatar
... mezmerizing

you made that in mathlab?
Ive heard lots ive good things about it... I only have mathematica...

Fav.
markdow's avatar
Yes, Matlab ("matrix lab"), which was derived from Fortran-like syntax. Designed for easy use with arrays (like images), instead of math operations (Mathmatica). But they are similar in many respects.

Enjoyed a lot of your algorithmically generated stuff.
Pinto-Warrior's avatar
thanks....

Yeah, its sounds like their functionality largely overlaps. So the images are rendered pointwize, wheras in mathematica they are generated based on primatives. btw, your avy is awesome! Ive been just staring at it for a while now....

Thanks. It takes a fair amount of thought to generate each image, which of course is true of any medium.
kidjet's avatar
djsmalley's avatar
Xeno-Minako's avatar
Whoa...very amazingly done. ^^
I love how there ARE a pair of lines that don't move at all, yet the other moves exactly perpendicular to it.
milamber55555's avatar
TRIPPY!!!!!!!!!!!!!! I LOVE IT!!!!!!!!!
gemstars's avatar
Urgh....I've been staring at this for too long! Far too long! I can see both ways, I've been trying to see how fast I can make it change back to the other way. Looking at it one way for a long time then saying 'Change!' ... the mind can be pretty stubborn, really. But it works. Wonderfully done. I liked your comment too, although your geeky explaination made my head go to goo... Well done, have a great day. :)
markdow's avatar
Hey, thanks for staring!

I've stared at it far too long also, but have found a trick that allows me to reliably switch percept: flicking my eyes between the center (rotating) and a fixed contour near the edge (fixed) induced the switch. Still I can't linger on one percept for more than several seconds before the other invades.
gemstars's avatar
Your little picture wave thing too...AHH! It's very different but...*Stares*

(You're welcome)
RUN415's avatar
Nice one Mark...intriguing.
djmathew's avatar
spyderraver's avatar
I like, I like. The artist's comments are great, too.
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